Suppose that $3\triangle_4=\triangle2_{11}$, where $\triangle$ represents a digit 0-9. What is the value of the missing digit?
Solution: After converting everything to base 10, we are able to solve for $\triangle$. We get \begin{align*}
3\triangle_4&=\triangle2_{11}\quad\Rightarrow\\
3\cdot4^1+\triangle\cdot4^0&=\triangle\cdot11^1+2\cdot11^0\quad\Rightarrow\\
12+\triangle&=11\cdot\triangle+2\quad\Rightarrow\\
10&=10\cdot\triangle\quad\Rightarrow\\
\boxed{1}&=\triangle.
\end{align*}